CN114647962A - Low-frequency elastic metamaterial high-order topological insulator and application - Google Patents
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Abstract
The invention belongs to the development of condensed state physics in the technical field of vibration control, and discloses a high-order topological insulator of a low-frequency elastic metamaterial and application thereof, wherein the high-order topological insulator consists of a first topological structure and a second topological structure which are different; an interface formed by the first topological structure and the second topological structure is a topological insulator boundary, and an included angle formed by the first topological structure and the second topological structure is a topological insulator corner; the first topological structure and the second topological structure are both formed by a plurality of cell periodic arrangements. The invention can realize the precise waveguide and fixed point local effect of low-frequency elastic waves, and the structure of the invention comprises a local resonance unit. When the structural parameter optimization is carried out by combining a genetic algorithm and the like, the precise regulation and control of the elastic wave in the depth sub-wavelength region can be realized, namely, the ratio of the lattice constant of the cellular structure in the working frequency region to the half wavelength of the elastic wave at the regulated and controlled specified frequency is less than 1, and the working frequency range and the size of the miniaturized structure are effectively reduced.
Description
Technical Field
The invention belongs to the development of condensed state physics in the technical field of vibration control, and particularly relates to a low-frequency elastic metamaterial high-order topological insulator and application thereof.
Background
At present, the acoustic topological insulator is widely concerned by scholars at home and abroad due to the excellent wave regulation and control capability of the acoustic topological insulator. The elastic high-order topological insulator can achieve the multi-dimensional regulation effect of elastic waves, and theories and experiments confirm that the waveguide effect of the elastic waves along a specified path or the wave convergence effect at a specified position can be achieved through a structural boundary or a corner formed by combining cellular structures with different topological characteristics.
Although many researches on elastic high-order topological insulators are currently carried out, and great research results are obtained in the analysis of the physical mechanism of the acoustic topology, most researches are carried out based on the elastic phononic crystal structure, and the following defects mainly exist: firstly, the traditional acoustic topological insulator mainly focuses on the high-frequency range for regulating and controlling the sound waves/elastic waves, and the problem of low-frequency regulation and control is difficult to solve; secondly, the traditional acoustic topology insulator has limited wave constraint capability and weak waveguide localization capability, and a large waveguide channel is often required to be constructed to realize good constraint. Therefore, there is an urgent need to develop a method for regulating and controlling low-frequency elastic waves or a method for designing a structure so as to overcome the defects caused by the bragg scattering cell design and fully exert the novel wave regulation and control capability of the topological insulator.
Compared with the traditional acoustic topological insulator, the high-order topological insulator realized by combining the low-frequency elastic metamaterial has superior characteristics in wave regulation. For example, a local resonance system is used for realizing a high-order topological insulator, which can realize elastic wave regulation and control in a sub-wavelength scale, so that the structure size is far smaller than the controlled elastic wave wavelength, and the size of a sample piece can be further miniaturized when low-frequency elastic wave regulation and control is carried out; more importantly, the localized resonance system enables the vibration energy to be localized in the vibrator while reducing the presence of vibration on the substrate, which has natural advantages for achieving highly energy localized topologies. Therefore, the design of the low-frequency elastic metamaterial high-order topological insulator is of great significance.
Through the above analysis, the problems and defects of the prior art are as follows:
(1) the traditional acoustic topology insulator mainly focuses on the high-frequency range for regulating and controlling the sound waves/elastic waves, and the problem of low-frequency regulation and control is difficult to solve.
(2) The traditional acoustic topology insulator has limited wave constraint capability and weak waveguide localization, and a large waveguide channel is often required to be constructed to realize good constraint.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a low-frequency elastic metamaterial high-order topological insulator and application thereof.
The invention is realized in such a way that the low-frequency elastic metamaterial high-order topological insulator is composed of two different topological structures, namely a first topological structure and a second topological structure:
an interface formed by the first topological structure and the second topological structure is a topological insulator boundary, and an included angle formed by the first topological structure and the second topological structure is a topological insulator corner; the first topological structure and the second topological structure are both formed by a plurality of cell periodic arrangements.
Further, the cells of the first topology and the second topology have different topology invariant characterizations, and the cells are respectively plain topology invariant cells or non-plain topology invariant cells.
Further, the plain topologically invariant cells constitute a finite triangular structure, and a bulk band gap for wave isolation can be realized.
Further, the non-trivial topologically invariant cells constitute a finite triangular structure, and boundary and angular states can be realized within the bulk bandgap.
Furthermore, each cellular unit consists of a regular hexagonal substrate and m local resonance units, and the thickness of the substrate is h; the distance between the centers of the adjacent cells is a lattice constant a; the m local resonance units form a cantilever beam with the width w and a circular support base with the radius ri through a C-shaped hole with the hollowed-out outer diameter ro of the substrate, and a mass block with the same radius and the height of hr is attached to the support base.
Furthermore, the tray bottom centers are spaced from the cell center d1 and distributed at equal intervals around the cell center to form a rotational symmetric structure, and the distance from the vertex is d 2.
Further, when the distance d1 between the bottom support center and the cell center is kept to be the maximum, the opening direction of the C-shaped hole is adjusted to form cantilever beams pointing to the inside or outside cells respectively. Meanwhile, the geometric position of the local resonance unit is described by the parameter beta (d1-d 2)/2.
Further, the base plate and the mass block are made of non-metal materials such as polyurethane and nylon or metal materials such as aluminum and copper, and the base plate and the mass block can be made of different materials.
Further, the substrate and the mass block are integrally printed and manufactured by a 3D printing technology or manufactured in a wire cutting machine processing mode; the mass block is fixed on the substrate through screws or directly adhered on the substrate.
The invention further aims to provide an application of the low-frequency elastic metamaterial high-order topological insulator in vibration control by using a condensed state physical related concept as a reference.
By combining the technical scheme and the technical problem to be solved, the technical scheme to be protected in the invention has the following advantages and positive effects:
firstly, aiming at the technical problems existing in the prior art and the difficulty in solving the problems, the technical scheme to be protected and the results, data and the like in the research and development process are closely combined, and some creative technical effects brought by the technical scheme are verified. The specific description is as follows:
the invention has the low-frequency elastic wave control performance. Compared with the traditional elastic topological insulator material, the elastic wave control device can realize the manipulation of the elastic wave in the depth sub-wavelength region range, namely the ratio of the lattice constant of the cellular structure to the half wavelength of the elastic wave at the regulated and controlled specified frequency is far less than 1. Specifically, as described in the embodiment, the wave regulation and control effect of 400-600Hz can be realized, the advantages of the local resonance metamaterial are fully exerted, and the miniaturization of the device is favorably realized.
The invention has good elastic wave regulation and localization capability. The elastic metamaterial high-order topological insulator can realize fixed-point local energy and waveguide along a specified path, and can reduce vibration of a substrate to a great extent and restrain energy on a structural corner or boundary.
The invention has the characteristics of simple structure and easy processing and manufacturing. The structural unit in the invention has simple shape and is convenient for machining. In addition, the structure can be selectively adjusted according to specific application scenarios by adjusting relevant parameters of the vibrator in the structure, such as the width, length, and height of the mass, so as to reach a specific target.
Secondly, considering the technical scheme as a whole or from the perspective of products, the technical effect and advantages of the technical scheme to be protected by the invention are specifically described as follows:
the invention can realize the precise waveguide and fixed point local effect of the low-frequency elastic wave, the structure of the invention comprises a local resonance unit, the parameter optimization of the structure is carried out by combining a genetic algorithm and the like, the precise regulation and control of the elastic wave in a deep sub-wavelength region can be realized, the working frequency range is effectively reduced, and the size of a miniaturized structure is effectively reduced.
Third, as inventive supplementary proof of the claims of the present invention, the following important aspects are also presented:
aiming at the difficult problem of fine regulation and control of elastic waves in a low-frequency range, the invention can realize high-efficiency regulation and control of the low-frequency elastic waves by fusing topological physics and local resonance structure design ideas, so that the ratio of the lattice constant of the cellular structure to the half wavelength of the elastic waves at the regulated and controlled specified frequency is far less than 1, and the development of integrated acoustic devices is facilitated. In addition, the invention fully exerts the elastic wave constraint advantage of the local resonance unit and has excellent energy localization capability.
Drawings
Fig. 1 is a schematic structural diagram of a low-frequency elastic metamaterial high-order topological insulator provided by an embodiment of the present invention.
Fig. 2 is a schematic diagram of an energy band structure of a unit cell and two unit cells at different oscillator positions according to an embodiment of the present invention;
in fig. 2: fig. a, β -0.24 and β -0.24 cellular schematic diagrams; and b, an energy band structure diagram obtained by performing energy band scanning on the two-element cells at different oscillator positions along a path of the first Brillouin zone 'gamma-K-M-gamma'.
FIG. 3 is a diagram illustrating a finite structure of a cell and its eigenstates according to an embodiment of the present invention;
in fig. 3: fig. a, a finite structure formed by 0.24 unit cells and eigenstates of the finite structure; and b, a finite structure formed by beta-0.24 unit cells and the eigenstate condition of the finite structure.
Fig. 4 is a schematic diagram of displacement field distribution of the high-order topological insulator provided by the embodiment of the present invention to realize elastic wave fixed-point local area when f is 541 Hz.
Fig. 5 is a schematic diagram of the displacement field distribution of the high-order topological insulator provided by the embodiment of the invention, which realizes the transmission of elastic waves along the topological insulator when f is 490 Hz.
In the figure: 1. a first topology; 2. a second topology; 3. a cellular cell; 4. a topological insulator boundary; 5. and (6) a corner.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not limit the invention.
First, an embodiment is explained. This section is an explanatory embodiment expanding on the claims so as to fully understand how the present invention is embodied by those skilled in the art.
As shown in fig. 1, the low-frequency elastic metamaterial high-order topological insulator provided by the embodiment of the present invention is composed of two different topological structures, namely a first topological structure 1 and a second topological structure 2: the interface formed by the first topology 1 and the second topology 2 is a topological insulator boundary 4, and the included angle formed by the first topology 1 and the second topology 2 is a corner 5. The first topological structure 1 and the second topological structure 2 are both formed by a plurality of cells 3 which are periodically arranged, and the cells 3 of the first topological structure 1 and the second topological structure 2 have different topological invariant (body polarization) representations.
Each cellular 3 is composed of a regular hexagonal substrate and m local resonance units, wherein the thickness of the substrate is h; the distance between the centers of the adjacent unit cells 3 is a lattice constant a; the m local resonance units form a cantilever beam with the width w and a circular support base with the radius ri through a C-shaped hole with the hollowed-out outer diameter ro of the substrate, the center of the support base is away from the center d1 of the cell and is distributed around the center of the cell at equal intervals to form a rotational symmetric structure, and the distance from the vertex is d 2; a mass with the same radius and height of hr is added on the circular support base. When the distance d1 between the bottom supporting center and the cell center is adjusted, in order to keep the band gap of the cell to be the maximum, the opening direction of the C-shaped hole can be adjusted to form cells with cantilever beams respectively pointing to the inside or the outside of the cell, and the geometric position of the local resonance unit is described by using the parameter β ═ d1-d 2)/2. The base plate and the mass block can be made of non-metal materials such as polyurethane and nylon, or metal materials such as aluminum and copper, and the base plate and the mass block can be made of different materials. The substrate and the mass block can be integrally printed and manufactured by a 3D printing technology, and can also be manufactured by machining modes such as wire cutting and the like; the mass block can be fixed on the substrate by screws or the like, or can be directly adhered on the substrate.
The cells 3 can realize topology peaceful or topology non-peaceful cells through the adjustment of the cell utilization parameter beta. Its topological properties can be characterized according to bulk polarization calculations and appear as: the limited triangular structure is formed by the plain topological invariant cells 3, and the body band gap of wave insulation can be realized; the finite triangular structure is formed by the cells 3 which are not mediocre topologically invariant, boundary states and angular states can be realized within the bulk bandgap.
In the low-frequency elastic metamaterial high-order topological insulator provided by the embodiment of the invention, the first topological structure and the second topological structure with different topological characteristics form the boundary and the corner of the topological insulator. When excitation with the frequency within the range of the cellular band gap is applied to the topological insulator, a boundary state and an angle state can be excited respectively, wherein the boundary state is represented as that elastic waves are transmitted along the boundary waveguide of the flapping insulator, and vibration is isolated along the inside of the structure body, and backscattering caused by defects such as immune bending, disorder and the like can be realized, so that the topological insulator has good robustness; the angular regime appears as an elastic wave localized efficiently on the corners of the flapping insulator, without spreading around the structure. By utilizing the characteristics of the boundary state and the angle state, the fine regulation and control of the elastic wave can be realized.
And II, application embodiment. In order to prove the creativity and the technical value of the technical scheme of the invention, the part is the application example of the technical scheme of the claims on specific products or related technologies.
As shown in fig. 2, the two different unit cells 3 of the first topology 1 and the second topology 2 provided by the embodiment of the present invention have structures that both satisfy C3 symmetry, the positions of the oscillators satisfy β ═ 0.24 and β ═ 0.24, respectively, and the directions of the cantilever beams are adjusted so as to make the band gap ranges of the two unit cells consistent. The thickness h of a substrate in the cellular is 1mm, the lattice constant a is 50mm, the cellular contains 3 vibrators, the hollow outer diameter ro is 6mm, the width w of a cantilever beam is 1mm, the inner diameter ri is 4mm, and the height hr of a mass block is 5 mm. The cellular substrate is made of aluminum, the material parameters are that the Young modulus is 70Gpa, the Poisson ratio is 0.33, and the density is 2700kg/m3The additional mass block is made of copper, the parameters of the material are Young modulus of 110Gpa, Poisson ratio of 0.35 and density of 8690kg/m3The proof mass is directly adhered to the substrate.
And simulating through a structural mechanics module in finite element software COMSOL, and introducing the calculated energy band structure into software MATLAB to obtain a dispersion curve graph in the figure 2. Although the binary cells have different topological invariant characteristics, similar band structures and band gaps of the energy band can be obtained by scanning the energy band along the path of the first Brillouin zone 'gamma-K-M-gamma', the band gap ranges from 466Hz to 573Hz, and the ratio of the structural lattice constant to the half wavelength ranges from 0.7 to 0.77, which indicates that the band gap ranges in the deep sub-wavelength region.
As shown in fig. 3, the embodiment of the present invention provides a finite structure with different topology invariants to characterize the cell composition. The structural eigenstates are simulated through a structural mechanics module in finite element software COMSOL, and the boundary of the structure is set as a low-reflection boundary condition during calculation. A finite triangular structure composed of topologically mediocre cells 3, a bulk band gap enabling wave insulation; whereas a finite triangular structure consisting of topologically non-mediocre cells 3 was found to be able to achieve boundary states and angular states within the body band gap, wherein the boundary states enable energy to propagate along the boundary and to be suppressed in the direction inside the structure, and the angular states enable energy to be localized at the structure corners and suppressed in the remaining directions, this property verifies that the two cells have different topological properties.
Further, the two kinds of cells with different topological characteristics are respectively used to form the low-frequency elastic metamaterial high-order topological insulator (as shown in fig. 1) provided by the embodiment of the invention, so that vibration control can be realized within the band gap range of the cells: the effect of elastic waves along the boundary waveguide can be realized in the boundary state frequency range, and the fixed point local effect of energy at the corner of the structure can be realized in the angular state frequency range.
And thirdly, evidence of relevant effects of the embodiment. The embodiment of the invention achieves some positive effects in the process of research and development or use, and has great advantages compared with the prior art, and the following contents are described by combining data, diagrams and the like in the test process.
The low-frequency elastic metamaterial high-order topological insulator in the figure 1 is formed by using two kinds of unit cells with different topological properties in the figure 2. On the basis of meeting the conditions, simulation is carried out through a response solver of a structural mechanics module in finite element software COMSOL, the structure is set to be a free boundary condition, excitation points are arranged at four corners of the structure, and the structure is excited.
When the excitation frequency is the angular state frequency of the topological insulator, as shown in fig. 4, the energy is almost completely localized on the oscillator at the corner 5 through the steady-state displacement field distribution condition, and high-stability local constraint of the energy is realized.
When the excitation frequency is the boundary state frequency of the topological insulator, the energy is constrained to propagate along the boundary 4 by the steady-state displacement field distribution condition, and the directional guide propagation of the energy is realized, as shown in fig. 5.
The elastic wave regulation and control frequency range realized by the high-order topological insulator provided by the invention is lower than 600Hz, and the ratio of the structural lattice constant to the half wavelength is smaller than 0.77, which indicates that the band gap range is positioned in a depth sub-wavelength region. Meanwhile, the structure shows a good energy localization effect, the vibration of the substrate can be greatly reduced, the energy is efficiently constrained on the corner or the boundary of the structure on a small-scale periodic structure, and the excellent characteristics of the embodiment of the invention are reflected.
The above description is only for the purpose of illustrating the present invention and the appended claims are not to be construed as limiting the scope of the invention, which is intended to cover all modifications, equivalents and improvements that are within the spirit and scope of the invention as defined by the appended claims.
Claims (10)
1. The low-frequency elastic metamaterial high-order topological insulator is characterized by comprising two different topological structures, namely a first topological structure and a second topological structure:
an interface formed by the first topological structure and the second topological structure is a topological insulator boundary, and an included angle formed by the first topological structure and the second topological structure is a topological insulator corner; the first topological structure and the second topological structure are both formed by a plurality of cell periodic arrangements.
2. The low frequency elastic metamaterial high order topological insulator of claim 1, wherein the cells of the first and second topologies have different topologically invariant characterizations, the cells being either plain topologically invariant or non-plain topologically invariant cells, respectively.
3. The low frequency elastic metamaterial high-order topological insulator of claim 2, wherein the unit cells with the plain topological invariants form a finite triangular structure, and a bulk band gap for wave insulation can be realized.
4. The low frequency elastic metamaterial high-order topological insulator of claim 2, wherein the cells that are not mediocre topological invariant form a finite triangular structure, and boundary states and angular states can be realized in a bulk band gap.
5. The low-frequency elastic metamaterial high-order topological insulator as claimed in claim 1, wherein the unit cells are each composed of a regular hexagonal substrate and m local resonance units, and the thickness of the substrate is h; the distance between the centers of the adjacent cells is a lattice constant a; the m local resonance units form a cantilever beam with the width w and a circular support base with the radius ri through a C-shaped hole with the hollowed-out outer diameter ro of the substrate, and a mass block with the same radius and the height of hr is attached to the support base.
6. The low frequency elastic metamaterial high-order topological insulator as claimed in claim 5, wherein the bottom supporting centers are spaced apart from the cell center d1 and distributed around the cell center at equal intervals to form a rotational symmetric structure, and the distance between the top supporting centers is d 2.
7. The low-frequency elastic metamaterial high-order topological insulator as claimed in claim 5, wherein when the distance between the bottom supporting center and the cell center is d1, in order to keep the band gap of the cell to be the maximum, the opening direction of the C-shaped hole is adjusted to form cells with cantilever beams respectively pointing to the inside or the outside of the cell, and the cell describes the geometric position of the local resonance unit by using the parameter β (d1-d 2)/2.
8. The low frequency elastic metamaterial high-order topological insulator as claimed in claim 5, wherein the base plate and the proof mass are made of polyurethane, nylon non-metal material or aluminum and copper metal material, and the base plate and the proof mass can be made of different materials.
9. The low-frequency elastic metamaterial high-order topological insulator as claimed in claim 5, wherein the substrate and the mass block are integrally printed and manufactured by a 3D printing technology or manufactured by a wire cutting machine; the mass block is fixed on the substrate through screws or directly adhered on the substrate.
10. Use of a low frequency elastic metamaterial high-order topological insulator designed by using condensed physical concepts as claimed in any one of claims 1 to 9 in vibration control.
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| CN116206589A (en) * | 2023-04-28 | 2023-06-02 | 南京南大电子智慧型服务机器人研究院有限公司 | Topological sonic wave guide with adjustable working frequency range |
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